A140404 -id:A140404 - cp's OEIS frontend

A050982 5-idempotent numbers.

1, 30, 525, 7000, 78750, 787500, 7218750, 61875000, 502734375, 3910156250, 29326171875, 213281250000, 1510742187500, 10458984375000, 70971679687500, 473144531250000, 3105010986328125, 20091247558593750, 128360748291015625, 810699462890625000

Offset: 5

Eric W. Weisstein

Number of n-permutations of 6 objects: t,u,v,z,x, y with repetition allowed, containing exactly five u's. Example: a(6)=30 because we have uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu, uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu, uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu, uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu, uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu. - Zerinvary Lajos, Jun 16 2008

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43.

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
Eric Weisstein's World of Mathematics, Idempotent Number.
Index entries for linear recurrences with constant coefficients, signature (30,-375,2500,-9375,18750,-15625).

Cf. A001788, A036216, A040075, A050988, A050989, A000389, A054849, A036219, A045543, A036084, A140404, A000389, A054849, A036219, A045543, A036084, A140404.

[Binomial(n, 5)*5^(n-5): n in [5..25]]; // Vincenzo Librandi, Aug 12 2017

seq(binomial(n, 5)*5^(n-5), n=5..32); # Zerinvary Lajos, Jun 16 2008

CoefficientList[Series[1 / (1 - 5 x)^6, {x, 0, 33}], x] (* Vincenzo Librandi, Aug 12 2017 *)

a(n)=binomial(n, 5)*5^(n-5) \\ Charles R Greathouse IV, Sep 03 2011

a(n) = C(n, 5)*5^(n-5).
G.f.: x^5/(1-5*x)^6. - Zerinvary Lajos, Aug 06 2008
From Amiram Eldar, Apr 17 2022: (Start)
Sum_{n>=5} 1/a(n) = 6400*log(5/4) - 17125/12.
Sum_{n>=5} (-1)^(n+1)/a(n) = 32400*log(6/5) - 23625/4. (End)

A170932 a(n) = binomial(n + 8, 8)*7^n .

Original entry on oeis.org

1, 63, 2205, 56595, 1188495, 21630609, 353299947, 5299499205, 74192988870, 980996186170, 12360551945742, 149450309889426, 1743586948709970, 19715944727720430, 216875392004924730, 2327795874186192102, 24441856678955017071, 251607348165713411025

Offset: 0

Zerinvary Lajos, Feb 08 2010

With a different offset, number of n-permutations of 8 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight, (8) u's.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (63,-1764,28812,-302526,2117682,-9882516,29647548,-51883209,40353607).

Cf. A027474, A140107, A139641, A140404, A036226, A050989.

[Binomial(n + 8, 8)*7^n: n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Table[Binomial[n + 8, 8]*7^n, {n, 0, 20}]

a(n) = C(n + 8, 8)*7^n.
From Amiram Eldar, Aug 29 2022: (Start)
Sum_{n>=0} 1/a(n) = 12082656/5 - 15676416*log(7/6).
Sum_{n>=0} (-1)^n/a(n) = 117440512*log(8/7) - 235229912/15. (End)

A197192 a(n) = binomial(n+9, 9)*7^n.

Original entry on oeis.org

1, 70, 2695, 75460, 1716715, 33647614, 588833245, 9421331920, 140142312310, 1961992372340, 26094498552122, 332111799754280, 4068369546989930, 48194531556649940, 554237112901474310, 6207455664496512272, 67894046330430602975

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A139641, A140107, A140404, A170932

Magma
```
[Binomial(n+9, 9)*7^n: n in [0..20]];
```

Table[Binomial[n+9,9]7^n,{n,0,20}] (* Harvey P. Dale, Jul 10 2025 *)

a(n) = C(n + 9, 9)*7^n.

A197193 a(n) = binomial(n+10, 10)*7^n.

Original entry on oeis.org

1, 77, 3234, 98098, 2403401, 50471421, 942133192, 16016264264, 252256162158, 3727785507446, 52188997104244, 697434779483988, 8950413003377846, 110847422580294862, 1330169070963538344, 15518639161241280680, 176524520459119567735, 1962537315692564605995, 21369850770874592376390, 228319984551975908021430

Offset: 0

Vincenzo Librandi, Oct 13 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (77,-2695,56595,-792330,7764834,-54353838,271769190,-951192165,2219448385,-3107227739,1977326743).

Cf. A139641, A140107, A140404, A170932

Magma
```
[Binomial(n+10, 10)*7^n: n in [0..20]];
```

Table[Binomial[n+10,10]7^n,{n,0,30}] (* or *) LinearRecurrence[{77,-2695,56595,-792330,7764834,-54353838,271769190,-951192165,2219448385,-3107227739,1977326743},{1,77,3234,98098,2403401,50471421,942133192,16016264264,252256162158,3727785507446,52188997104244},30] (* Harvey P. Dale, Jul 11 2025 *)

a(n) = C(n + 10, 10)*7^n.

G.f.: -1 / (7*x-1)^11 . - R. J. Mathar, Oct 13 2011

cp's OEIS Frontend

A050982 5-idempotent numbers.

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A170932 a(n) = binomial(n + 8, 8)*7^n .

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Magma

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A197192 a(n) = binomial(n+9, 9)*7^n.

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Author

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Programs

Magma

Mathematica

Formula

A197193 a(n) = binomial(n+10, 10)*7^n.

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Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula